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Theorem ereldm 8697
Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ereldm.1 (𝜑𝑅 Er 𝑋)
ereldm.2 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Assertion
Ref Expression
ereldm (𝜑 → (𝐴𝑋𝐵𝑋))

Proof of Theorem ereldm
StepHypRef Expression
1 ereldm.2 . . . 4 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
21neeq1d 3004 . . 3 (𝜑 → ([𝐴]𝑅 ≠ ∅ ↔ [𝐵]𝑅 ≠ ∅))
3 ecdmn0 8696 . . 3 (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
4 ecdmn0 8696 . . 3 (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅)
52, 3, 43bitr4g 314 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐵 ∈ dom 𝑅))
6 ereldm.1 . . . 4 (𝜑𝑅 Er 𝑋)
7 erdm 8659 . . . 4 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
86, 7syl 17 . . 3 (𝜑 → dom 𝑅 = 𝑋)
98eleq2d 2824 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐴𝑋))
108eleq2d 2824 . 2 (𝜑 → (𝐵 ∈ dom 𝑅𝐵𝑋))
115, 9, 103bitr3d 309 1 (𝜑 → (𝐴𝑋𝐵𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  wne 2944  c0 4283  dom cdm 5634   Er wer 8646  [cec 8647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-er 8649  df-ec 8651
This theorem is referenced by:  erth  8698  brecop  8750  eceqoveq  8762
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